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Research Article | Open Access

Optimal boundary regularity for mixed local and nonlocal equations

Nicola Abatangelo1( ), Elisa Affili2, Matteo Cozzi3
Dipartimento di Matematica, Alma Mater Studiorum Università di Bologna, P.zza di Porta S. Donato 5, 40126 Bologna, Italy
Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen Normandie, Avenue de l'Université BP.12, 76801 Saint-Étienne-du-Rouvray, France
Dipartimento di Matematica "Federigo Enriques", Università degli Studi di Milano, via Saldini 50, 20133 Milan, Italy
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Abstract

We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach makes use of weighted Hölder spaces as well as regularity estimates for the Laplacian in this context and a fixed-point argument. We show the optimality of the obtained estimates by means of a counterexample that we have striven to keep as explicit as possible.

Keywords

elliptic regularitymixed operatorsweighted Hölder spacesboundary value problemsnonlocal operatorsfractional Laplacian

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Mathematics in Engineering
Volume 8 Issue 1,
February 2026
Pages 1-42
DOI: 10.3934/mine.2026001

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Cite this article:
Abatangelo N, Affili E, Cozzi M. Optimal boundary regularity for mixed local and nonlocal equations. Mathematics in Engineering, 2026, 8(1): 1-42. https://doi.org/10.3934/mine.2026001

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Received: 21 July 2025
Revised: 09 December 2025
Accepted: 23 December 2025
Published: 15 February 2026
©2026 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)